Non-unique solutions for electron MHD

We consider the electron magnetohydrodynamics (MHD) equation on the 3D torus ${\mathbb{T}}^3$. For a given smooth vector field $H$ with zero mean and zero divergence, we can construct a weak solution $B$ to the electron MHD in the space $L_t^{\gamma }{W}_x^{1,p}$ for appropriate $(\gamma ,p)$ such that $B$ is arbitrarily close to $H$ in this space. The parameters $\gamma$ and $p$ depend on the resistivity. As a consequence, non-uniqueness of weak solutions is obtained for the electron MHD with hyper-resistivity. In particular, non-Leray-Hopf solutions can be constructed. As a byproduct, we also show the existence of weak solutions to the electron MHD without resistivity.